Let X be a normed linear space such that its dual X' is separable, then show that X is separable ? Is the converse true ? Justify your answer.

Q: 3. Show that the set W = {(x, 0) : x e R} is a subspace of R?.

A: In order to show W is a subspace of R2 , we need to prove W is closed under addition. For any two…

Q: Let Y be a vector subspace of a normed space X. Show that the closure of Y is also a vector…

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Q: State if the following statement is true or false: If y is a vector space of dimension , then any…

A: Given:

Q: If V is a vector space than show that cV=V

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Q: Let Y = {(x,y,z)| z=-x}. (a) Is Y a subspace of R? Justify. (b) Is Y a vector space? Justify.

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Q: If X, is a dense subspace of a normed linear space X show that X X'.

A: We will use basic knowledge of dual of normed linear spaces and the concept of isometric…

Q: Show if W is a subspace of the vector space X, then W is convex.

A: Definition used: A set K in a vector space V is called convex if and only if  x, y belongs to K,…

Q: What is the dimension of the vector space S=abc detEM2x3(R): a=b=c=0}

A: Dimension of vector space is number of element in basis

Q: Let V = {(x, y): x 20) with standard operation on R2. Is V a vector space?

A: Given set is  V={(x,y):x=>0 }

Q: True or False: If U and W are subspaces of a finite dimensional vector space and UnW = {0}, then U°…

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Q: If V is a finite dimensional vector space and W is a subspace, the W is finite dimensional. Prove it

A: NOTE:  If V is a finite dimensional vector space and W is a subspace of V, then W is finite…

Q: 6. Show that the closure Ỹ of a subspace Y of a normed space X is again a vector subspace.

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Q: 10. Let Y = {(x,y,z)| z=-x}. (a) Is Y a subspace of R³? Justify. (b) Is Y a vector space? Justify.

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Q: Let V be an n-dimensional vector space with inner product ( , ). Let U and W be subspaces of V such…

A: Let V be an finite dimensional inner product space and U and W be subspace of V and U∩W=0¯ We have…

Q: Determine if the set W of vectors (X1 , X2) in R such that x1² + x2 = 0 is a subspace of R?.

A: We have to determine whether W is a subspace of R^2 or not.

Q: Let S be the space of all complex vectors (x, y, z) such that y = z². Determine whether or not S is…

A: In the given problem we verify the properties of vector addition

Q: Let Y be a closed subspace of a normed linear space X. Show that X is complete X are complete. Y AY…

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Q: et X be a normed linear space and T: X→ X and S: X→ X be two perators, then S T :X→X is a bounded…

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Q: Determine whether W is a subspace of V. V = F W is the set of all integrable functions

A: Given V = F is a vector space of function Also given W is the set of all integrable functions

Q: Let S={x | x=a+b +c 1 where a,b,c,deR. Show explicitly that S is a vector space.

A: We will construct a transformation matrix whose span will be the set S, and the  we will show that…

Q: Show that {t, e'}, which is a subset of F, the vector space containing all real-valued functions, is…

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Q: Determine whether the set A= (x,y,z):x+y%3D3Z is a subspace of R or not?

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Q: Let V ((x,y): x 20) with standard operation on R2. Is V a vector space?

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Q: Let-{ », :x, y, a, b e R and x - a = y – b}.Is W a subspace?

A: We will use the one step test for vector subspace to check if W is a subspace or not. We know that…

Q: If X a normed space and y is a proper proper Linear subspace of X. show that y has empty interior.

A: We proved by contradiction. Suppose Y has a nonempty interior then Y contains a open ball. Then we…

Q: Let V be a vector space over R, and dimV=200. Let NEL(V) be a nilpotent operator such that null(N2)…

A: Consider the given information.

Q: Determine the dimension of the vector space. R5

A: If V is a part of a finite set, then V is said to be finite-dimensional, and the dimension of V,…

Q: Let V = {(x, y): x 20} with standard operation on R². Is V a vector %3D space?

A: Our objective is to conclude V is a vector space or not.

Q: Show that if a non-zero element in a normed space X, then there is a bounded linear functional f on…

A: We need to show that if a non-zero element in a normed space X, then there is a bounded linear…

Q: If M = {x: Lx = 0}, where L is a continuous linear functional on H, prove that M¹ is a vector space…

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Q: Let V = {(x, y): x 20} with standard operation on R2. Is V a vector space?

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Q: Give a counter example to show that W is not subspace of R³. W=set of (x, y, z) where x + y = 1

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Q: Let V = {(x,y) where y= 3x+1 x ER} with scalar multiplication defined by k(x,y)=(kx,y) is a vector…

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Q: Verify that the set C[a, b] of all continuous real-valued functions defined on the interval a ≤ x ≤…

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Q: Give a counter example to show that W is not subspace of R³. W=set of (x, y, z) where x + y = 1 6. ,

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Q: 4. Consider C" as a real vector space. It has a subspace U := C x {0}"-1. What is the dimension of…

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Q: (b). Let V = R? = {(r,,1)| plication. Is it a vector space? IER} with usual addition and scalar…

A: Is it vector space

Q: ement at time t, a, and az are arbitrary constants, and o is a fixed constant. Determine if the set…

A: Given, the mass in a mass-spring is pulled downward and then released, causing the system to…

Q: W = {(x, y, z): x 2 0},

A: Given the vector space

Q: Let V be a finite-dimensional vector space and let U be a subspace of V. Show that if Lim(U) =…

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Q: Let Wi and W2 be two two-dimensional subspaces of the linear space R³, that is dim W1 W = W1n W2 is…

A: Given W1 and W2be the two-dimensional subspace of linear space ℝ3. And dim W1=dim W2=2 let  x∈W1∩W2…

Q: a) Prove that W = {(a,b, 2a + b)} is a subspace of R. (more space on the next %3D if needed)

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Q: Let X be a normed linear space which is non-zero. Prove that X is a Banach space if and only if the…

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Q: Let V and W be subspaces of R", and dim V and dim W are strictly less than n. Assume that no…

A: To prove: If V and W are subspaces of ℝn such that dim V < n, dim W < n, V is not contained in…

Q: Let V = R and W = {x|x = (x1, x2, 1)}, Is W a subspace of V?

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Q: If W is a subspace of the vector space V = R, then the vector zero belongs to W: Select one: O True

A: A subset W of a vector space V is called a subspace of V if W is itself a vector space under the…

Q: Verify that the set C[a, b] of all continuous real-valued functions defined on the interval a ≤ x ≤…

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Q: Show that if W is a subspace of a finite-dimensional vec- tor space V, then W is finite-dimensional…

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Q: Let T be a linear operator on the finite – dimensional vector space

A: